Exponential mixing for the fractional Magneto-Hydrodynamic equations with degenerate stochastic forcing

Ergodicity and exponential mixing of the real Ginzburg-Landau equation with a degenerate noiss

In this paper, we establish the existence, uniqueness and attraction properties of an invariant measure for the real Ginzburg-Landau equation in the presence of a degenerate stochastic forcing acting only in four directions. The main challenge is to establish time asymptotic smoothing properties of the Markovian dynamics corresponding to this system. To achieve this, we propose a condition which only requires four noises

经典无限晶格上具有高度退化噪声的自守和周期测度

Invariant and periodic measures in classical spin systems on infinite lattices with highly degenerate noise

本文研究了无限维随机微分方程描述的经典自旋系统，在仅对一个粒子施加随机力的情况下，证明了弱马尔可夫解的存在性、唯一性和马尔可夫性质，并展示了噪声在自旋粒子中的传播，从而得到唯一自守/周期测度和几何混合性。

Ergodicity of the 2D Navier-Stokes Equations with Degenerate Stochastic Forcing

The stochastic 2D Navier-Stokes equations on the torus driven by degenerate noise are studied. We characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic. In particular, our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in $Ł^2_0(\TT^2)$. Unlike previous works, this class is independent of the viscosity and the strength of the noise. The two main tools of our analysis are the \textit{asymptotic strong Feller} property, introduced in this work, and an approxima…

Uniqueness of the Invariant Measure for a Stochastic PDE Driven by Degenerate Noise

We consider the stochastic Ginzburg-Landau equation in a bounded domain. We assume the stochastic forcing acts only on high spatial frequencies. The low-lying frequencies are then only connected to this forcing through the non-linear (cubic) term of the Ginzburg-Landau equation. Under these assumptions, we show that the stochastic PDE has a unique invariant measure. The techniques of proof combine a controllability argument for the low-lying frequencies with an infinite dimensional version of the Malliavin calculus to show positivity and regularity of the invariant measure. This then implies t…

Landau-Lifshitz-Gilbert equations: Controllability by Low Modes Forcing for deterministic version and Support Theorems for Stochastic version

In this article, we study the controllability issues of the Landau-Lifshitz-Gilbert Equations (LLGEs), accompanied with non-zero exchange energy only, in an interval in one spatial dimension with Neumann boundary conditions. The paper is of twofold. In the first part of the paper, we study the controllability issues of the LLGEs. The control force acting here is degenerate i.e., it acts through a few numbers of low mode frequencies. We exploit the Fourier series expansion of the solution. We borrow methods of differential geometric control theory (Lie bracket generating property) to establish …

退化纯跳跃噪声驱动的分数阶磁流体动力学方程的遍历性

Ergodicity for the fractional Magneto-Hydrodynamic equations driven by a degenerate pure jump noise

本文研究了由高度退化纯跳跃 Lévy 噪声驱动的随机 2D 分数阶磁流体动力学方程的遍历性，针对噪声仅作用于四个方向的高退化情形，结合 Malliavin 变分法证明半群 e-property 与弱不可约性，确立了不变测度的唯一性。

Invariant Measures for Nonlinear Conservation Laws Driven by Stochastic Forcing

Some recent developments in the analysis of long-time behaviors of stochastic solutions of nonlinear conservation laws driven by stochastic forcing are surveyed. The existence and uniqueness of invariant measures are established for anisotropic degenerate parabolic-hyperbolic conservation laws of second-order driven by white noises. Some further developments, problems, and challenges in this direction are also discussed.

Exponential Mixing for a Stochastic PDE Driven by Degenerate Noise

We study stochastic partial differential equations of the reaction-diffusion type. We show that, even if the forcing is very degenerate (i.e. has not full rank), one has exponential convergence towards the invariant measure. The convergence takes place in the topology induced by a weighted variation norm and uses a kind of (uniform) Doeblin condition.

Viscous scalar conservation law with stochastic forcing: strong solution and invariant measure

We are interested in viscous scalar conservation laws with a white-in-time but spatially correlated stochastic forcing. The equation is assumed to be one-dimensional and periodic in the space variable, and its flux function to be locally Lipschitz continuous and have at most polynomial growth. Neither the flux nor the noise need to be non-degenerate. In a first part, we show the existence and uniqueness of a global solution in a strong sense. In a second part, we establish the existence and uniqueness of an invariant measure for this strong solution.

Ergodicity of the finite dimensional approximation of the 3D Navier-Stokes equations forced by a degenerate noise

We prove ergodicity of the finite dimensional approximations of the three dimensional Navier-Stokes equations, driven by a random force. The forcing noise acts only on a few modes and some algebraic conditions on the forced modes are found that imply the ergodicity. The convergence rate to the unique invariant measure is shown to be exponential.

具有退化噪声的随机偏微分方程的大偏差极小化子

Large Deviation Minimisers for Stochastic Partial Differential Equations with Degenerate Noise

针对多稳态与退化噪声共存导致标准方法难以求解的问题，本文结合最优控制、大偏差理论与数值优化计算大偏差极小化子，并在生物、医学及流体动力学等应用中验证了方法效率。

底部局部随机强迫的Rayleigh-Bénard convection

Rayleigh-Bénard convection with stochastic forcing localised near the bottom

本文证明了在无限Prandtl数regime下，当随机扰动作用于底部薄层且noise的amplitude足够大时，三维Rayleigh-Bénard convection的随机流会稳定到唯一的stationary measure。

通过有色噪声越过鞍点：理论与数值计算

Escape over a saddle by coloured noise: theory and numerics

研究随机动力系统中越过鞍点的转变，通过哈密顿最优控制法、线性滤波器及划分方法（MOD）分别处理退化噪声、有色噪声和无限时间范围，以解决现有计算方法难题，通过实例展示了方法有效性。

Stationary measures for stochastic differential equations with degenerate damping

A variety of physical phenomena involve the nonlinear transfer of energy from weakly damped modes subjected to external forcing to other modes which are more heavily damped. In this work we explore this in (finite-dimensional) stochastic differential equations in $\mathbb R^n$ with a quadratic, conservative nonlinearity $B(x,x)$ and a linear damping term $-Ax$ which is degenerate in the sense that $\mathrm{ker} A \neq \emptyset$. We investigate sufficient conditions to deduce the existence of a stationary measure for the associated Markov semigroups. Existence of such measures is straightforwa…

随机理想二进磁流体动力学模型的唯一性

Uniqueness for a stochastic ideal dyadic MHD model

研究无粘性无电阻磁流体动力学随机二进模型，针对特定随机强迫证明弱唯一性，但解耗散形式上守恒的能量。